Proposition: Flags, conditioning, and the reflected-carrier example

Suppose the complete output has orthogonal flags $f$, with message-independent probabilities $p_f$, and normalized conditional ensembles ${\rho }_{x,f}$. Then

In particular, if a message is captured with probability ${\eta }_{\mathrm{cap}}$, its captured accessible branch is constant, and an intact reflected qubit is available on the other branch, then

Proof. The requested-bit differences are block diagonal in $f$. Their trace norms are sums of the block trace norms, proving (eq:flagaverage). Apply the constant-branch score $1/2$ and the intact-qubit score $q_{\mathrm{Q}}$ to obtain (eq:reflectionfalsepositive).